Question

1. Let x and y represent two animal populations which satisfy (9-1-3y) y(-6 + 2x) (a) (5 points) What is the relationship between x and y? How does a grow in the absence of y? How does y grow in the absence of x? (b) (5 points) Sketch the nullclines and direction arrows of the system. (c) (4 points) Find the eigenvalues of the interior critical point.

Answer 1

o civen system is. da dt - re (9-1-by) 위분위 dy at = y (-6 tare) a In the above system term ny denotes the relationship between a and y. in first equation equation there is -32%. shows decreases due to y. In the second equation there is any it shows 9 increases due prey as Predator. The > to re, Hence cond y

relationship is prey - predator relationship. nullclines. r(9-x-3y) y (-6+27) direction of systen. (0,3) (3,2) 2 |(3,0) remaining part of ca) at if y = 0 (absence of y) du alg-n) grows logistically it ( absence of a) dy -64 =

y will die out after some time. interior equilibrium in solution tollowing equations. g-x-3y =o -6+an - O → 2-3 putting 3 in first equation. 6 - 39 Y=2 (3,2) so, interior equilibrium is. Jacobian matrix ts. .9-22-39 -32 JE 24 - 6+22 at point (3,2) 3 9 J 13,2) = 4 O characteristic equation es. +² +32 +36 2- - 3+ 9-144 -3 3.15 i 2 2

4 General solution. y 4 3 1 2 co move in region 1, dk >o, dy dt at so in this region trajectories downwand and right in region 2, du dy 70, > 0 dt at upward and right so trajectories move dn in region 3, <o dy ct >o at upward and left. so trajectories move in region 4, da <o, at dy at dwn ward & Left trajectories move

(3,2) stable critical point y (1,5) 16 (3,2)